Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equations ẋ=f(x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g , i.e. varieties which are controlled invariant sets of ẋ=f(x)+g(x)u for any choice of the drift vector f . We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV’s, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.